My goal is to add a discrete variable of $X = \{-1,0,1\}$, to a uniform variable $Z \in [-1,1]$ and to have a new uniform variable.
If I design $p_X = [0.5, 0, 0.5]$, does this hold? My hesitation is that $0$ will come from both $-1+1$ and $1 - 1$, and at this point we will have a bit more density than the rest.
The same probability measure can have different probability densities, but the probability densities are allowed to be different up to a measure zero set. In your case, the density that you compute found is different from the uniform density only at a single point. Therefore it is fine in the sense that, for every function, $g, \int_\mathbb{R}g(x)f(x)dx = \int_\mathbb{R}g(x)\hat f(x)dx$, f is the uniform density and $\hat f$ is the density that you compute.